138 XIII INTEGRATION
Now the sets Apt9tf are closed (3.11.4), hence BM?r is integrable (13.8.7). More-
over, by hypothesis, for each integer r ^ 1 we have ^/f) Bw>r\ = 0, and the
sequence (B^Xg! is decreasing, so that by (13.8.7) we have lim yu(Bn>r) = 0.
Hence there exists an integer nr such that n(Bnrtf) g s/2r+2. Let B = (J BMr,r;
the set B is integrable, and we have //(B) g f] s/2r+2 = Je by (13.8.7). The
set C = K0 - B is also integrable, and by definition the sequence (fn \ C)
converges uniformly to f\ C. Hence we may take K' to be any compact
subset of C such that n(C - K') ^ & (13.7.9).
(13.9.11) Let (fn) be any sequence of measurable functions on X with values in
R. Then the functions inffn> sup/M, lim sup/M and lim inffn are measurable.
n n ./i-* oo /i-* oo
For sup/, is the limit of the increasing sequence of functions gn = sup ft,
which are measurable (13.9.7); and lim sup/n is the limit of the decreasing
sequence of functions sup/n+p, which are measurable, by what has just been
(220.127.116.11) The propositions (13.9.6), (13.9.7), (13.9.9), and (13.9.11) remain
valid when the word "measurable" is replaced throughout by "universally
measurable." On the other hand, a function cannot be universally measurable
unless it is everywhere defined, because any nonempty subset N of X has
measure //(N) ^ 0 if for example \JL is the Dirac measure at a point of N.
In particular (12.7.8), every function belonging to«/ (resp. ff1) is universally
measurable. If / is any lower semicontinuous function on X, then by
/ = inf (sup (/, - n)} and sup (/, - n) = ~ n + sup (/ + n, 0)
and using (13.9.6), it follows that
(13L9.11.2) Every (upper or lower) semicontinuous function on X with values
in R is universally measurable.
The same will be true of all functions obtained from semi-continuous
functions by repeated applications of the operations in (13.9.6), (13.9.7), and
(13.9.11).ictions u \ Ln is continuous.